THE 1990 ASIAN PACIFIC MATHEMATICAL OLYMPIADTime allowed: 4 hours NO calculators are to be used.. For each value of ∠BAC, how many non-similar triangles are there in which AEGF is a cycl
Trang 1THE 1990 ASIAN PACIFIC MATHEMATICAL OLYMPIAD
Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.
Question 1
Given triagnle ABC, let D, E, F be the midpoints of BC, AC, AB respectively and let G
be the centroid of the triangle
For each value of ∠BAC, how many non-similar triangles are there in which AEGF is a
cyclic quadrilateral?
Question 2
Let a1, a2, , a n be positive real numbers, and let S k be the sum of the products of a1, a2,
, a n taken k at a time Show that
S k S n−k ≥
µ
n k
¶2
a1a2· · · a n for k = 1, 2, , n − 1.
Question 3
Consider all the triangles ABC which have a fixed base AB and whose altitude from C is a constant h For which of these triangles is the product of its altitudes a maximum?
Question 4
A set of 1990 persons is divided into non-intersecting subsets in such a way that
1 No one in a subset knows all the others in the subset,
2 Among any three persons in a subset, there are always at least two who do not know each other, and
3 For any two persons in a subset who do not know each other, there is exactly one person
in the same subset knowing both of them
(a) Prove that within each subset, every person has the same number of acquaintances (b) Determine the maximum possible number of subsets
Note: It is understood that if a person A knows person B, then person B will know person A; an acquaintance is someone who is known Every person is assumed to know one’s self.
Question 5
Show that for every integer n ≥ 6, there exists a convex hexagon which can be dissected into exactly n congruent triangles.